Abstract
We classify all the \(n\)-component links in the \(3\)-sphere that bound a Thurston norm minimizing Seifert surface \(\Sigma\) with Euler characteristic \(\chi(\Sigma)=n-2\) and that are nearly fibered, which means that the rank of their link Floer homology group \(\widehat{HFL}\) in the maximal (collapsed) Alexander grading \(s_{\text{top}}\) is equal to two. In other words, such a link \(L\) satisfies \(s_{\text{top}}=\frac{n-\chi(\Sigma)}{2}=1\), and in addition \({\rm rk}\widehat{HFL}_{*}(L)[1]=2\) and \({\rm rk}\widehat{HFL}_{*}(L)[s]=0\) for every \(s>1\).
The proof of the main theorem is inspired by the one of a similar recent result for knots by Baldwin and Sivek, and involves techniques from sutured Floer homology. Furthermore, we also compute the group \(\widehat{HFL}\) for each of these links.
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Acknowledgement
We thank Steven Sivek for a helpful conversation, and the referee for their corrections.
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AC has a CIRGET post-doctoral fellowship at the Université du Québec à Montréal. IM is supported by the Knut and Alice Wallenberg Foundation through the grant KAW 2021.0191, and by the Swedish Research Counsil through the grant number 2020-04426, as a postdoctoral researcher at Uppsala Universitet. We are both grateful to the Alfréd Rényi Institute of Mathematics in Budapest for its hospitality during the semester on "Singularities and low-dimensional topology", and for the support from the Élvonal (Frontier) grant KKP126683 (given by NKFIH).
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Cavallo, A., Matkovič, I. Nearly fibered links with genus one. Acta Math. Hungar. 170, 721–731 (2023). https://doi.org/10.1007/s10474-023-01364-0
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DOI: https://doi.org/10.1007/s10474-023-01364-0